Extremes of the two-dimensional Gaussian free field with scale-dependent variance

Arguin, Louis-Pierre; Ouimet, Frédéric

doi:10.48550/arxiv.1508.06253

Cited by 3 publications

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“…( 70) is positive and log-convex (in the interval containing t = 0 where it has no poles) if Z t is; had we chosen the naïve continuation of ( 58), this would no longer be the case since we would have instead " exp(tF ) = Γ(1−tβ −1 ) Γ(1−t) Z −t e tc ′′ . Now, it is easy to see that the prediction for the fluctuating part of the free energy F encoded in (70), which was obtained solely from arguments based on 1RSB and analytical continuations, is equivalent to the freezing scenario (25): Indeed, the RHS of (25) (for β > 1) and ( 70) coincide. By (52), the LHS side of ( 70) equals e tβ −1 ln Z0 R −∂ y G β (y)e ty dy, which is just (25), provided we make the identification of F (M, β > 1, f ij ) in ( 26) with the unknown and formally divergent constant in (69):…”

Section: Freezing Scenariomentioning

confidence: 96%

“…),(26) is the non-fluctuating part of the minimum, and contributes a deterministic shift to ln Z[ρ]. Up to translation, − ln Z[ρ] has the same distribution as the critical temperature free energy, as one can see from the above equation and(70). Note that up to now in the present paper the over-line in Z t (11) was a formal symbol and did not correspond to any genuine average, so (E20) gives it a probability interpretation, in terms of the random measure ρ.…”

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confidence: 93%

“…Note that this correct analytical continuation leads to the factor Γ(1 + t/β) in the l.h.s. of(70), which is the Laplace transform of the PDF of −G/β, where G is a standard Gumbel random variable.…”

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confidence: 99%

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One step replica symmetry breaking and extreme order statistics of logarithmic REMs

2016

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Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ( "infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits.In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of 1/f -noise.regardless of the geometric nature of the model. In fact eq. (4) holds for both uncorrelated REM and logREM that satisfy (2), and a way to see this for uncorrelated REM is as follows [1]. By (2), one computes the microscopic ensemble entropy, defined as the log of the number of states up to a given energy: S(E = −2q ln M ) = ln M (1 − q) for q ∈ [0, 1] and S = 0 for q > 1, and notices the non-analyticity at q = 1 where β c = ∂S ∂E q=1−ǫ = 1. For correlated REMs, strictly speaking, the existence of one (and only one) transition is a result of RSB analysis, see Sect. IV A and A.

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Section: Freezing Scenariomentioning

confidence: 96%

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confidence: 93%

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One step replica symmetry breaking and extreme order statistics of logarithmic REMs

2016

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“…However the property of duality invariance is not restricted to the replica limit n = 0. In [12] by considering the generating functions for the moments of the partition function, one notices that duality invariance can be extended to finite n by further requiring n β = nβ (see (25) there where s should be identified as −nβ).…”

Section: Dualitymentioning

confidence: 99%

“…In particular, the problem of characterizing the distribution of the global maximum M N = max x∈D N V N,x of LCG fields and processes (or their continuum analogues) recently attracted a lot of interest, in physics, see [6,11,12,13,14,15] and mathematics, see [16,17,18,2,19,20,21,22,23,24,25,26]. The distribution is proved to be given by the Gumbel distribution with random shift [20] and has a universal tail predicted by renormalization group arguments in [6].…”

Section: Introductionmentioning

confidence: 99%

Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

2016

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We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index H → 0 (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the freezing-duality conjecture (FDC). Here we study the PDF of the position of the maximum x m through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the β-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary β > 0 and positive integer n in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix 1 from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit n → 0 and to negative Dyson index β → −2, we obtain the moments of x m and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.With Appendix 1 written

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Extrema of Log-correlated Random Variables: Principles and Examples

Arguin

2016

Advances in Disordered Systems, Random Processes and Some Applications

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Abstract. These notes were written for the mini-course Extrema of log-correlated random variables: Principles and Examples at the Introductory School held in January 2015 at the Centre International de Rencontres Mathématiques in Marseille. There have been many advances in the understanding of the high values of log-correlated random fields from the physics and mathematics perspectives in recent years. These fields admit correlations that decay approximately like the logarithm of the inverse of the distance between index points. Examples include branching random walks and the twodimensional Gaussian free field. In this paper, we review the properties of such fields and survey the progress in describing the statistics of their extremes. The branching random walk is used as a guiding example to prove the correct leading and subleading order of the maximum following the multiscale refinement of the second moment method of Kistler. The approach sheds light on a conjecture of Fyodorov, Hiary & Keating on the maximum of the Riemann zeta function on an interval of the critical line and of the characteristic polynomial of random unitary matrices.

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Extremes of the two-dimensional Gaussian free field with scale-dependent variance

Cited by 3 publications

References 0 publications

One step replica symmetry breaking and extreme order statistics of logarithmic REMs

One step replica symmetry breaking and extreme order statistics of logarithmic REMs

Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes

Extrema of Log-correlated Random Variables: Principles and Examples

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